Compactness theorems of gradient Ricci solitons

Abstract

In this paper, we prove the compactness theorem for gradient Ricci solitons. Let $(M_{\alpha },g_{\alpha })$ be a sequence of compact gradient Ricci solitons of dimension $n\ge 4$, whose curvatures have uniformly bounded $L^{\frac{n}{2}}$ norms, whose Ricci curvatures are uniformly bounded from below with uniformly lower bounded volume and with uniformly upper bounded diameter, then there must exists a subsequence $(M_{\alpha },g_{\alpha })$ converging to a compact orbifold $(M_{\infty },g_{\infty })$ with finitly many isolated singularities, where $g_{\infty }$ is a gradient Ricci soliton metric in an orbifold sense.

21pages